Method For Determining Time to Failure of Submicron Metal Interconnects

ABSTRACT

The present disclosure is related to a method for determining time to failure characteristics of a microelectronics device. A test structure, being a parallel connection of a plurality of such on-chip interconnects, is provided. Measurements are performed on the test structure under test conditions for current density and temperature. The test structure is arranged such that failure of one of the on-chip interconnects within the parallel connection changes the test conditions for at least one of the other individual on-chip interconnects of the parallel connection. From these measurements, time to failure characteristics are determined, whereby the change in the test conditions is compensated for.

BACKGROUND

The present invention is related to a method for determining thelifetime characteristics of submicron metal interconnects.

In the microelectronics industry scaling refers to the miniaturisationof active components and connections on a chip. It has followed Moore'slaw for many decades. Despite many advantages, scaling also stronglyinfluences the reliability and time to failure of interconnects, i.e.the (metal) conductor lines connecting elements of the integratedcircuit. Electromigration (EM), i.e. the mass transport of a metal dueto the momentum transfer between conducting electrons and diffusingmetal ions, is one of the most severe failure mechanisms of on-chipinterconnects. A major problem when testing the reliability of newcomponents is that their time to failure under real life conditions(T_(max)=125° C.; j=1-3 10⁵ A/cm² for a standard type IC) is alwaysextremely long (in order of years). For that reason, the physicalfailure mechanisms are studied and methods are established foraccelerating these mechanisms. The failure times of the devices inoperation are measured and models are developed for extrapolating theseresults to real life conditions. The electromigration acceleratingconditions for these tests are the temperature T and the current densityj. Reliability tests are usually performed on identical interconnects ataccelerating conditions (j,T) (170° C.<T<350° C. and 10⁶<j<10⁷ A/cm²instead of T=125° C. and j=1-3.10⁵ A/cm²). All interconnects are eachindividually connected with an own power supply and provided with amultiplexer. Moreover, each interconnect can be found on a different ICpackage, which is an expensive and time-consuming activity. In order toderive the activation energy, which is a parameter describing thetemperature dependence of the observed degradation, these tests must beperformed at three temperatures, therefore tripling the number of powersupplies, multiplexers and IC packages. This makes the tests morecomplex and expensive. Nevertheless these reliability tests are of greatimportance to manufacturers because on the one hand continuouslyoperational IC's are indispensable and on the other hand the competitivestrength of manufacturers strongly depends on the reliability of theirproducts. Therefore such tests should provide a large amount ofstatistical information in a relatively short period of time, whilekeeping costs under control. It is hard to lower the costs withoutdecreasing the accuracy of the experiments. To solve this problem oneneed to look for a test structure that can provide a large amount ofaccurate data in a short period of time at low cost.

Patent document US-2002/0017906-A1 discloses a method for detectingearly failures in a large ensemble of semiconductor elements. It employsa parallel test structure. A Wheatstone bridge arrangement is used tomeasure small resistance changes. The criterion for failure of the teststructure is the time to first discernible voltage imbalance ΔV(t).

SUMMARY

The present disclosure aims to provide a method for determining the timeto failure that allows manufacturers to perform reliability tests in arelatively fast and inexpensive way.

The disclosure relates to a method for determining time to failurecharacteristics of an on-chip interconnect subject to electromigration.One such method includes the steps of

-   -   providing a test structure, being a parallel connection of a        plurality of such on-chip interconnects,    -   performing measurements on the test structure under test        conditions for current density and temperature, the test        structure being such that failure of one of the on-chip        interconnects within the parallel connection changes the test        conditions for at least one of the other individual on-chip        interconnects of the parallel connection,    -   determining from the measurements estimates of the time to        failure characteristics, whereby the change in the test        conditions is compensated for.

Preferably the step of determining is based on fitting. Advantageouslythe current density exponent n and the activation energy E_(a) aredetermined with the fitting.

The time to failure characteristics preferably include the median timeto failure of the on-chip interconnects as well as the shape parameter.

In a preferred embodiment the method further comprises a step ofcorrecting the shape parameter estimation, the shape parameter beingdetermined by fitting. The step of correcting the shape parameter isadvantageously performed via a predetermined relationship.

Preferably the parallel connection of the on-chip interconnects iswithin a single package.

In an alternative embodiment the method further comprises the step ofperforming measurements on the individual devices, belonging to theparallel connection. The step of compensating typically includescorrecting the measurements on the individual interconnects.

The measurements advantageously are performed with electromigrationaccelerating test conditions for current density and temperature.

In an another advantageous embodiment the determining step includes thestep of determining the time to failure characteristics under real lifeconditions by extrapolation. The extrapolation typically uses the Blackmodel, which is used to describe the temperature and current dependencyof the observed degradation.

The measurements are advantageously resistance change measurements.

In a specific embodiment the electromigration acceleratering testconditions are used as input values.

Further the number of interconnects within parallel connection may beused as input value. Also the failure criterion can be used as input.

Preferably the measurements are performed at several time instances.Advantageously the measurements are performed at least three differenttemperatures.

In a preferred embodiment the on-chip interconnect is in a 90 nmtechnology. Alternatively it is in a sub90 nm technology.

The invention also relates to a program, executable on a programmabledevice containing instructions, which when executed, perform the methodas described before.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a parallel test structure as used inthe method of the present invention.

FIG. 2 is a graph of the relative drift for 3 series and parallelinterconnects as a function of time for n=2.

FIG. 3 is a graph of the current through 3 series and 3 parallelinterconnects as a function of time.

FIG. 4 is a graph of σ_(par/σ and σ) _(ser)/σ as a function of n forseveral input values of σ (0.1 till 1.0) and with a fixed FC=1, whereσ_(par) and σ_(ser) are respectively the shape parameter for theparallel and series interconnects gained from the simulationexperiments. σ is the input parameter for the simulation experiments.

FIG. 5 is a graph of σ_(par)/σ and σ_(ser)/σ as a function of n forσ=0.5 and with FC varying from 0.5 to 3.

FIG. 6 is a graph of σ_(par)/σ as a function of FC for σ=0.5 and with nvarying from 1 to 20.

FIG. 7 is a graph of the cumulative failure plot of the failure times ofthe parallel interconnects from simulation 9 in table III.

FIG. 8 represents a block diagram of a Median Time to Failure testsystem.

FIG. 9 represents a scheme of a gastube oven.

FIG. 10 is a graph of the evolution of the interconnect resistance andthe oven temperature (T_(i)=180° C., I=25 mA) (FIG. 10 a) and a currentprofile of the test (FIG. 10 b).

FIG. 11 a is a graph of the evolution of the resistance and FIG. 11 bthe temperature versus time during a test (I_(AC)=10 mA, T_(s)=100° C.).

FIG. 12 is a graph of the current profile, resistance and oventemperature change versus time of the MTF test system (left) and theconventional system (right) for determining the current density exponentn.

DETAILED DESCRIPTION

This specification discloses a method for determining time to failurecharacteristics of a metal interconnect using a test structurecomprising a plurality of such interconnects. From a statistical pointof view, a series connection of metal interconnects is the optimumconfiguration for the reliability tests, because the same current passesthrough all the series connected metal lines. However, a series teststructure may be difficult to use in practice due to technicallimitations of the measurement equipment.

Alternatively, a test structure with a parallel configuration is used(see FIG. 1). Such a structure yields much faster reliability data tothe customer than conventional structures thanks to their inherentlylarger statistical information. Besides the obtained reduction inmeasurement time it will furthermore be possible for the customer toapply less accelerating test conditions (i.e. test conditions closer toreal-life). This renders the extrapolation of test data to userconditions more straightforward and reliable. As an additionaladvantage, the approach described herein allows use of only one currentsource for the whole test structure instead of one current source perinterconnect.

The method described herein is validated by mathematical simulation. Inorder to calculate and to compare the behaviour of parallel and/orseries test structures, simulation experiments have been carried out andfurther analysed by means of both the total resistance (TR) analysis anda software package for reliability data analysis. The former method usesthe resistance of the global structure (being series or parallelconnected) and is therefore called total resistance method. Thestructure which is constituted of a set of individual interconnects istreated as one structure. The resistance behaviour of this globalstructure is monitored and the time at which the failure criterion isexceeded is determined as can be seen in Table I, TR analysis. Thesecond method called ‘reliability data analysis’ makes use of thebehaviour of each individual interconnect that is part of the globalstructure which can be a parallel or series connection of the individualinterconnects. For each individual interconnect, the time to reach thefailure criterion has been determined. A cumulative failure distributioncan be derived using this information and by means of a commerciallyavailable software package the distribution parameters (μ and σ) can bedetermined. In practice, only the TR-analysis can be used for the seriesor parallel test structures.

The parameter studied during the accelerating conditions is the relativeresistance change, defined as $\begin{matrix}{\frac{\Delta\quad{R(t)}}{R_{0}} \equiv \frac{{R(t)} - {R\left( {t = 0} \right)}}{R\left( {t = 0} \right)}} & \left( {{equation}\quad 1} \right)\end{matrix}$The resistance R(t) is the resistance of the interconnect at time t andat accelerating conditions j (current density) and T (temperature). Itcan be shown that ΔR(t)/R₀ changes linearly as a function of time. FC isdefined as the failure criterion of an interconnect. This means thatinterconnects with a drift ΔR(t)/R₀ exceeding FC, are considered asfailed. Subsequently, $\begin{matrix}{\frac{\Delta\quad{R(t)}}{R_{0}} = {\frac{FC}{t_{F}}t}} & \left( {{equation}\quad 2} \right)\end{matrix}$where t_(F) is the failure time of the interconnect. The studiedquantity is the median of the failure times, i.e. the time where 50% ofthe interconnects failed, according to the failure criterion.

For extrapolation of the simulation results to more real lifeconditions, the Black-model is used, which is the most intensively usedextrapolation model. This model relates the median time to failureMT_(F) of a set of interconnects with the temperature T (in K) and thecurrent density j (in MA/cm²). $\begin{matrix}{{MT}_{F} = {{Aj}^{- n}{\exp\left( \frac{E_{a}}{k_{B}T} \right)}}} & \left( {{equation}\quad 3} \right)\end{matrix}$where A is a material constant, k_(B) the Boltzmann constant, E_(a) theactivation energy (in eV) of the thermally driven process and n thecurrent density exponent, which usually has a value between 1 and 3.

For a typical test a set of N interconnects is taken with acceleratingconditions j and T. It is assumed that both j and T are constant andthat the failure times of the interconnects obey to a monomodaldistribution. Moreover, a lognormal distribution is taken, because it isthe far most commonly used distribution, i.e. t_(f)∝ log n(ν,σ). So, thenatural logarithm of t_(f), ln(t_(f)), has a normal distribution.Moreover, its mean μ is given by the natural logarithm of the mediantime to failure. The median time to failure and σ are called the scaleand shape parameter of the distributed failure times, respectively.Given the 1-to-1 relationship between the mean μ of a lognormaldistribution and the median time to failure MT_(F), further on thenotation log n (MT_(F), σ) is applied, which clearly is to beinterpreted as log n (μ=ln(MT_(F)), σ). Due to transformations ofstatistical distributions, the relative resistance ΔR is lognormallydistributed, ΔR ∝ log n (ΔR_(med)′,σ), with median ΔR_(med)′ given by$\begin{matrix}{{\Delta\quad R_{med}^{\prime}} = \frac{R_{0}{FC}}{{MT}_{F}}} & \left( {{equation}\quad 4} \right)\end{matrix}$The shape parameter σ is the same as for the failure times. At higheraccelerating conditions (different j or T), the resistance change perunit of time is also lognormally distributed. Using the Black equation(3), the scale parameter ΔR_(med)′ (j,T) at higher acceleratingconditions can be written as $\begin{matrix}{{\Delta\quad{R_{med}^{\prime}\left( {j,T} \right)}} = {\Delta\quad{R_{med}^{\prime}\left( {j_{1},T_{1}} \right)}\left( \frac{j_{1}}{j} \right)^{n}{\exp\left\lbrack {\frac{E_{a}}{k_{B}}\left( {\frac{1}{T} - \frac{1}{T_{1}}} \right)} \right\rbrack}}} & \left( {{equation}\quad 5} \right)\end{matrix}$where ΔR_(med)′ (j₁,T₁) is the scale parameter of the lognormallydistributed ΔR at accelerating conditions j₁ and T₁ and the shapeparameter is the same for all accelerating conditions.

The simulation of a series electromigration test structure of N parallelinterconnects is quite simple. For this structure, the current density jin equation (3) is assumed constant as a function of time and as aconsequence, for each interconnection, the resistance change per unit oftime DR_(i)(t)=DR_(i)(1) is also constant as a function of time.Moreover, DR_(i)(t)∝ log n(ΔR_(med)′, σ). Taking for simplicityR_(i)(t=0) for interconnect i at a constant level R₀, ∀i, the resistancefor each interconnection per unit of time is given byR _(i)(t)=R ₀ +t·DR _(i)(1)  (equation 6)Using equations (2) and (6), the relative resistance change for thetotal structure is $\begin{matrix}{\frac{\Delta\quad{R_{tot}(t)}}{R_{tot}(0)} = {{\frac{\sum\limits_{i = 1}^{N}\quad{{DR}_{i}(1)}}{R_{tot}(0)} \cdot t} = {{\frac{\sum\limits_{i = 1}^{N}\quad{{DR}_{i}(1)}}{N \cdot R_{0}} \cdot t} \cong {\frac{\mu^{\prime}}{R_{0}} \cdot t}}}} & \left( {{equation}\quad 7} \right)\end{matrix}$where μ′ is the mean of the lognormally distributed DR_(i)(1) values forthe N interconnects. Subsequently, the failure time of the total seriesstructure can be approximated by $\begin{matrix}{t_{F} \cong \frac{{FC} \cdot R_{0}}{\mu^{\prime}} \cong \frac{\Delta\quad{R_{med}^{\prime} \cdot {MT}_{F}}}{\mu^{\prime}}} & \left( {{equation}\quad 8} \right)\end{matrix}$

For parallel structures, where the current density j_(i) is notconstant, the situation is far more complicated. The resistance R_(i)and the current density j_(i) of the i^(th) interconnection after 1 timeunit t₁, respectively, are given by $\begin{matrix}{{R_{i}\left( t_{1} \right)} = {{R_{0}\frac{FC}{t_{f}}t_{1}} + R_{0}}} & \left( {{equation}\quad 9} \right) \\{{j_{i}\left( t_{1} \right)} = {\frac{j_{tot}}{R_{i}\left( t_{1} \right)}{R_{tot}\left( t_{1} \right)}}} & \left( {{equation}\quad 10} \right) \\{j_{tot} = {N \cdot j_{0}}} & \left( {{equation}\quad 11} \right)\end{matrix}$where i=1, . . . , N and j₀ denotes the mean current density throughevery interconnect at time 0, j_(tot) the constant current density ofthe structure and R_(i)(t₁) the resistance of the parallel structure attime t₁. At time t₂=2t₁, it is assumed that the current density does notchange during this step, so $\begin{matrix}{{R_{i}\left( t_{2} \right)} = {{R_{i}\left( t_{1} \right)} + {\left( {{R_{i}\left( t_{1} \right)} - R_{0}} \right)\left\lbrack \frac{j_{i}\left( t_{1} \right)}{j_{0}} \right\rbrack}^{n}}} & \left( {{equation}\quad 12} \right)\end{matrix}$In contrast to a series structure it is not easy to derive an equationfor the failure time for the total structure. Only by simulationexperiments it is possible to study the behaviour of the currents andrelative resistance changes of the individual interconnects.

The results of the simulation are analysed. The four estimatedparameters are MT_(F) and σ for t_(f)∝ log n (MT_(F),σ), the activationenergy E_(a) and the current density exponent n, respectively. Note thatone is primarily interested in E_(a) and n. For obtaining those valuesalso the MT_(F) need be determined. For the simulation experiments Ninterconnects in a series and/or parallel structure are taken withlength L=2000 μm, thickness d=0.5 μm, width b=0.5 μm, resistivityρ_(A1)=2.68 μΩcm, current density j₁=2MA/cm², temperature T₁=200° C.,current density exponent n=2, activation energy E_(a)=0.8 eV, failuretimes t_(f) ∝ log n (200, 0.5), failure criterion FC=1 and 500 timesteps of 1 hour as standard input values.

A. Relative Resistance Change for Series and Parallel Interconnects

For the series interconnects the relative resistance drift of theinterconnects increases linearly, while for the parallel interconnectsthe relative resistance drift of the individual interconnects bendstowards each other. FIG. 2 shows the relative drift DR/R for 3 seriesand parallel interconnects as a function of time. The drift velocity ofinterconnects that start with a relatively high resistance rise,decreases, while the drift velocity of the interconnects that start witha relatively low resistance rise, increases. This is because a lowercurrent will run through the interconnects with the highest rise ofresistance, decreasing the accelerating conditions for theseinterconnects and therefore making these interconnects less sensitivefor electromigration at this point in time. The current will thendistribute over the other interconnects, making these interconnects moresensitive to EM. In FIG. 3 the current is shown for the interconnectsfrom FIG. 2. Moreover, it appears that as the current density exponent nincreases, the drift of the individual interconnects bow more towardseach other.

B. Influence of the Current Density Exponent

Table I shows the scale parameters MT_(F) and shape parameters σestimations for 80 parallel and series interconnects as a function ofthe current density exponent n, using FC=1. TABLE I Reliability dataanalysis TR-Analyse Series Parallel Series parallel n = 1, 2, 3 n = 1 n= 2 n = 3 n = 1, 2, 3 n = 1 , 2, 3 sim MT_(F) σ MT_(F) σ MT_(F) σ MT_(F)σ Tf tf 1 193 0.522 191 0.403 191 0.317 191 0.256 168 196 2 187 0.464184 0.355 184 0.279 185 0.225 167 188 3 204 0.402 204 0.323 204 0.254204 0.204 169 207 4 202 0.453 200 0.358 200 0.283 201 0.229 180 205 5209 0.476 207 0.373 207 0.293 208 0.236 187 212 6 196 0.504 194 0.393192 0.262 195 0.252 172 199 7 203 0.437 202 0.339 203 0.274 203 0.222185 207 8 220 0.457 218 0.358 218 0.282 220 0.227 197 223 9 200 0.505198 0.394 198 0.31 199 0.25 176 203 10 202 0.44 202 0.35 202 0.275 2020.222 184 206 11 214 0.472 213 0.366 213 0.286 213 0.23 193 217 12 2090.481 209 0.38 209 0.3 209 0.242 187 214 13 196 0.491 195 0.384 1950.302 195 0.244 174 199 14 193 0.484 190 0.368 190 0.289 191 0.233 171194 15 186 0.469 185 0.367 185 0.295 186 0.237 167 189 16 194 0.454 1910.347 191 0.273 192 0.22 174 195 17 182 0.511 181 0.409 181 0.322 1820.26 159 186 18 203 0.399 201 0.302 201 0.237 202 0.191 188 204 19 2040.466 202 0.364 202 0.286 203 0.231 183 207 20 188 0.453 185 0.344 1850.27 186 0.217 169 189 mean 199 0.47 198 0.364 198 0.284 198 0.231 179202 error 9 0.032 7 0.026 6 0.021 5 0.016 10 10

It is to be noted that μ for both series and parallel interconnectsagree with the proposed median failure time of 200 hours of theindividual interconnects used as an input parameter for the simulateddata. Table I shows that for every current density exponent n, thefailure time of the total parallel structure (with FC=1) agrees withsaid value of 200 hours for the scale parameter MT_(F). This means thata parallel EM test structure is a correct approach for the determinationof the failure time of submicron interconnects. For the total seriesstructure, the failure time of the total structure deviates from theproposed value for MT_(F) of the individual interconnects. This caneasily be explained by the asymmetry of the distributed failure timesand using equations (7) and (8).

For series interconnects the shape parameter obtained with thereliability data analysis is in agreement with the expected value of0.5, while for parallel interconnects the shape parameter is smaller.This is due to the fact that for the parallel interconnects, therelative resistance drifts of the individual interconnects bow towardseach other, as mentioned before. For increasing n, the drifts of theparallel interconnects bend even more towards each other, therebydecreasing the shape parameter. Therefore further investigations on thedependence of the shape parameter σ on several input parameters arerequired.

The shape parameter obtained from the simulation experiments withreliability data analysis is denoted σ_(par) for the parallel structureand σ_(ser) for the serial structure. The shape parameter input valuefor the simulation experiments is denoted σ. FIG. 4 shows respectivelyσ_(par) and σ_(ser) divided by the input value σ for the simulationexperiments as a function of n for several values of σ and with a fixedfailure criterion FC=1. For the series structure σ_(ser)/σ isindependent of n for several values of σ. For the parallel structure afixed relationship of σ_(par)/σ as a function of n can be found forseveral values of σ. This means that for a fixed FC value, a certainrelationship of σ_(par)/σ as a function of n can be found, independentof the input value σ.

To determine the influence of the failure criterion FC on σ_(par)/σ as afunction of n, simulation experiments were carried out with a fixedσ=0.5 and with FC varying from 0.5 to 3. The results are shown in FIG.5. The results show that the shape of the curves for different FC valuesdoesn't change. For FC>1, the curves move more away from 1, while theopposite is true for FC<1. FIG. 6 shows the relationship betweenσ_(par)/σ and FC for different values of n. It is shown that σ_(par)/σdecreases in the same way, but that for different n, the relationshipshifts more towards or more away from 1.

As a consequence of these results, using both FIGS. 5 and 6, and knowingboth n and FC, it is possible to estimate the correct shape parameterfor parallel test structures. This information extracted from thesimulation experiments can be used to correct the shape parameter forreal time experiments. However, the method described above may be rathertedious, because every relationship between σ_(par)/σ and n must beknown for all FC values, which is a time-consuming activity. Moreover,the acquired corrected value for σ_(par) by applying this method is onlya rough estimation of the real value. The method further requires thatfor each interconnect info on R and I are to be kept. In order to get amore accurate value for the shape parameter, an alternative method isproposed for the correction of the experimental data gained from aparallel structure. More in particular, the alternative method willyield an improved estimate of the shape parameter.

Still a parallel structure of N interconnects is considered. At everymoment in time it is known thatΔV(t)=R _(j)(t)·I _(j)(t)  (equation 13)where R_(j)(t) and I_(j)(t) are the resistance and current value,respectively, of interconnect j at time t. Furthermore, $\begin{matrix}{{I_{tot}(t)} = {\Delta\quad{{V(t)} \cdot {\sum\limits_{j = 1}^{N}\quad{1/{R_{j}(t)}}}}}} & \left( {{equation}\quad 14} \right)\end{matrix}$Subsequently, $\begin{matrix}{{I_{j}(t)} = {\frac{\Delta\quad(t)}{R_{j}(t)} = \frac{I_{tot}(t)}{{R_{j}(t)} \cdot \left( {\sum\limits_{j = 1}^{N}\frac{1}{R_{j}(t)}} \right)}}} & \left( {{equation}\quad 15} \right)\end{matrix}$Let <1/R>_(t) denote the mean of the values 1/R_(j)(t), i.e.$\begin{matrix}{\left\langle \frac{1}{R} \right\rangle_{t} = {\frac{1}{N} \cdot {\sum\limits_{j = 1}^{N}\left( \frac{1}{R_{j}(t)} \right)}}} & \left( {{equation}\quad 16} \right)\end{matrix}$Moreover, if I₀=I_(tot)/N denotes the mean current per interconnect,equation (15) can be written as $\begin{matrix}{{I_{j}(t)} = \frac{1_{0}}{{R_{j}(t)} \cdot \left\langle \frac{1}{R} \right\rangle_{t}}} & \left( {{equation}\quad 17} \right)\end{matrix}$As already mentioned previously, ΔR(t)/R₀ changes linearly as a functionof time: $\begin{matrix}{{\frac{\Delta\quad{R(t)}}{R_{0}} \equiv \frac{{R(t)} - {R\left( {t = 0} \right)}}{R\left( {t = 0} \right)}} = {a \cdot t}} & \left( {{equation}\quad 18} \right)\end{matrix}$where a is a constant.Subsequently, for every interconnect j, one can write that$\begin{matrix}{{\frac{1}{R_{j\quad 0}} \cdot \frac{\mathbb{d}{R_{j}(t)}}{\mathbb{d}t}} = {a_{j} = \frac{FC}{\left( t_{F} \right)_{j}}}} & \left( {{equation}\quad 19} \right)\end{matrix}$Where FC and (t_(F))_(j) are the failure criterion and the failure timeof the j^(th) interconnect, respectively. Postulate a_(j)=a_(j)(t) as afunction of time t, temperature T and current I. In agreement with theBlack equation, one can then write $\begin{matrix}{{a_{j}(t)} = {a_{j\quad 0} \cdot \left\lbrack \frac{I_{j}(t)}{I_{0}} \right\rbrack^{n}}} & \left( {{equation}\quad 20} \right)\end{matrix}$a_(j0) is the coefficient for interconnect j, with current I=I₀ and acertain temperature T.a _(j0) =a _(j)(I=I ₀ ,T)  (equation 21)Combining equations (19) and (20) $\begin{matrix}{{\frac{1}{R_{j\quad 0}} \cdot \frac{\mathbb{d}{R_{j}(t)}}{\mathbb{d}t}} = {a_{\quad{j\quad 0}} \cdot \left\lbrack \frac{I_{j}(t)}{\quad_{I_{0}}} \right\rbrack^{n}}} & \left( {{equation}\quad 22} \right)\end{matrix}$Substitution of equation (17) in equation (22) gives $\begin{matrix}{{\frac{1}{R_{j\quad 0}} \cdot \frac{\mathbb{d}{R_{j}(t)}}{\mathbb{d}t}} = {a_{\quad{j\quad 0}} \cdot \left\lbrack \frac{1}{\quad{{R_{j}(t)} \cdot \left\langle \frac{1}{R} \right\rangle_{t}}} \right\rbrack^{n}}} & \left( {{equation}\quad 23} \right) \\{or} & \quad \\{{a_{j\quad 0} \cdot t} = {\frac{1}{R_{j\quad 0}} \cdot t \cdot \frac{\mathbb{d}{R_{j}(t)}}{\mathbb{d}t} \cdot \left\lbrack {{R_{j}(t)} \cdot \left\langle \frac{1}{R} \right\rangle_{i}} \right\rbrack^{n}}} & \left( {{equation}\quad 24} \right)\end{matrix}$With this equation, it is possible to recalculate the a_(j0) value. Inpractice, all R_(j)(t) are known and subsequently also the <1/R>_(t)values at every t. The only concern is to determine the derivative ofthe R_(j)(t) values. In practice, several techniques to determine thederivative of R_(j)(t) are available. If both R_(j)(t) and itsderivative are known, the right side of equation can be calculated forevery t value. If then a straight line is fitted through the correctedpoints, the slope of this straight line gives the a_(j0) value forinterconnect j. Because a_(j)=FC/(t_(f))_(j), the correct failure timescan be determined with the correct MT_(F) and σ.

The theoretical verification of this method can be achieved by applyingit to the data gained from simulation experiments on parallelstructures. The simulations are performed in exactly the same conditionsas for obtaining the experimental results of Table I. Table II shows theMT_(F) and σ values, which were estimated from the simulation data ofboth parallel and corrected structures. Note that the corrected datafrom the simulation experiments on parallel structures is in goodagreement with the expected values for MT_(F) (=200) and σ (=0.5), usedas input parameters for the simulation experiments. This shows thismethod can correct the data gained from parallel test structures. TABLEII Parallel Correction n = 1 n = 2 n = 3 n = 1 n = 2 n = 3 sim MT_(F) σMT_(F) σ MT_(F) σ MT_(F) σ MT_(F) σ MT_(F) σ 1 212 0.35 212 0.274 2130.221 214 0.454 214 0.454 214 0.454 2 211 0.373 211 0.294 212 0.237 2120.468 212 0.468 212 0.468 3 217 0.383 217 0.307 218 0.244 218 0.486 2180.486 218 0.486 4 202 0.367 202 0.289 203 0.233 205 0.479 205 0.479 2050.479 5 185 0.359 185 0.282 186 0.227 188 0.469 188 0.469 188 0.469 6208 0.392 208 0.308 209 0.248 210 0.501 210 0.501 210 0.501 7 188 0.393188 0.309 188 0.249 190 0.506 190 0.506 190 0.506 8 208 0.334 208 0.268209 0.216 208 0.424 208 0.424 208 0.424 9 212 0.374 212 0.295 213 0.238212 0.468 212 0.468 212 0.468 10 190 0.377 190 0.296 191 0.239 192 0.486192 0.486 192 0.486 11 179 0.369 179 0.291 180 0.235 181 0.471 181 0.471181 0.471 12 198 0.353 198 0.283 198 0.221 199 0.443 199 0.443 199 0.44313 196 0.422 196 0.334 197 0.269 197 0.532 197 0.532 197 0.532 14 1800.385 180 0.286 180 0.231 182 0.476 182 0.476 182 0.476 15 206 0.365 2060.287 207 0.231 209 0.741 209 0.741 209 0.741 16 189 0.409 189 0.321 1900.259 191 0.525 191 0.525 191 0.525 17 208 0.434 208 0.344 209 0.278 2110.55 211 0.55 211 0.55 18 197 0.394 197 0.310 197 0.250 198 0.501 1980.501 198 0.501 19 191 0.396 191 0.314 192 0.253 192 0.497 192 0.497 1920.497 20 184 0.343 184 0.269 184 0.216 186 0.455 186 0.455 186 0.455mean 198 0.379 198 0.298 199 0.240 200 0.497 200 0.497 200 0.497 error 80.026 6 0.022 5 0.018 10 0.033 10 0.033 10 0.033C. Determination of Activation Energies E_(a)

Consider a system driven by only 1 activation energy (E_(a)=0.8 eV),which in practice is mostly the case. Suppose the failure time t_(F) ofthe interconnects at accelerating conditions j₁=2MA/cm² and T₁=200° C.is lognormally distributed with MT_(F1)=200 and σ=0.5 using FC=1. Athigher temperatures 220° C. and 240° C., the scale parameters can becalculated using equation (7). Table III shows the activation energiescalculated via the Black equation for both series and parallel teststructures, using the TR-analysis and the software package for 80 seriesand parallel interconnects (n=2). TABLE III “Failure” analysisTR-analysis n = 2 n = 2 n = 2 n = 2 series parallel series Parallel SimE_(a) E_(a) E_(a) E_(a) 1 0.772 0.774 0.772 0.774 2 0.746 0.749 0.7610.745 3 0.876 0.884 0.894 0.881 4 0.757 0.759 0.743 0.761 5 0.817 0.8220.825 0.82 6 0.752 0.756 0.747 0.756 7 0.805 0.814 0.843 0.808 8 0.8150.821 0.831 0.818 9 0.802 0.806 0.806 0.806 10 0.805 0.814 0.831 0.81011 0.809 0.812 0.804 0.813 12 0.858 0.866 0.875 0.863 13 0.854 0.860.879 0.855 14 0.792 0.792 0.791 0.791 15 0.759 0.765 0.767 0.764 160.778 0.783 0.809 0.776 17 0.718 0.725 0.724 0.724 18 0.801 0.804 0.8270.799 19 0.793 0.796 0.790 0.796 20 0.778 0.781 0.803 0.776 mean 0.790.80 0.81 0.80 error 0.02 0.02 0.1 0.1

For both methods the mean values of the activation energies agree verywell the proposed value of 0.8 eV that was an input parameter for thesimulated data. This implies that for the monomodal lognormallydistributed failure times, both series and parallel test structures canbe used for the determination of both the activation energy and thecurrent density exponent n. Also n is computed via the Black equation.FIG. 7 shows the cumulative failure plot of the failure times for the 80parallel interconnects of simulation experiment 9 in tables I and III.

As already mentioned before, the current density and the temperature ofthe sample have to remain constant during the entire test in order toguarantee an accurate time to failure determination. To keep thetemperature of the sample constant during the entire test, the jouleheating of the interconnect has to be taken into account, Joule heatingis created when a current is applied to the interconnect. Theinterconnect temperature is given by the following equation:T _(i) =T _(o) +P _(Ei)θ_(i) with P_(Ei)=I_(DC) ²R_(i)  (equation 25)where T_(i) is the actual temperature of the interconnect, T_(o) theoven temperature, P_(Ei) the power dissipated by the interconnect, θ_(i)the thermal resistance and R_(i) the electrical resistance of eachinterconnect.

In conventional median time to failure (MT_(F)) tests the joule heatingof the interconnect is estimated and the oven temperature is decreasedwith the estimated value of the joule heating in the beginning of thetest, but this is not sufficient. During the electromigration (EM) testthe resistance of the interconnect increases and so the joule heatingalso increases. From equation 25 it can be concluded that theinterconnect temperature will increase, thus giving rise to theacceleration of the EM process.

This problem can be solved by using an AC current based dynamic joulecorrection, by which no EM damage occurs. During the determination ofthe thermal resistance of the interconnect the same temperature and thesame RMS value of the current is used as in the proper EM test. Thisapproach is described with more details in the following paragraphs.

A block diagram of the mean time to failure test system is shown in FIG.8. It consists of a PC, two measuring units (MU), a multiplexer (MUX), aprogrammable voltage source (VS), a switch box (SB), a programmablecurrent source (CS) and a thermal unit (TU). A GPIB-bus interconnectsthe PC, the MU's, the VS and the CS. The thermal unit consists of agastube oven with a sample holder and a Pt100-temperature sensorconnected flow upwards of the sample holder (FIG. 9). A gastube oven isespecially used because of several features. In a gastube oven differentatmospheres can be created. An atmosphere of helium can be used forexample. From the different measurements a place is determined where thePt100-temperature sensor picks up only 1% of the joule heating of thesample and where the temperature relationship T_(o)(T_(s)) is stable,T_(o) being the oven temperature and T_(s) the sample temperaturemeasured by the Pt100-temperature sensor.

For the EM test the test system sequentially performs the followingsteps: annealing of the sample and determination of the temperaturecoefficient of resistance (TCR), estimation of the thermal resistanceand the EM test with dynamic joule correction. During the first step(FIG. 10) annealing takes place, in which a small AC current is suppliedto the sample, until all defects are annealed. During the second stepthe thermal resistance of the interconnect is determined at exactly thesame sample temperature as during the actual EM test by supplying an ACcurrent with the same RMS value as the DC-current of the EM test. Byusing this AC current, the same joule heating is created as during theEM test, but no EM damage occurs. When the high AC current is applied,the resistance of the interconnect rises rapidly due to the jouleheating, as can be seen in FIG. 10. The oven temperature is then loweredbased on equation 25. A starting value has to be used for the thermalresistance, and a few iterations have to be performed to find theoptimum thermal resistance value. The iteration is complete when theresistance value at the end of phase 2 equals the value at the end ofphase 1 (FIG. 11). During the last step the EM test is performed with aconstant sample temperature by making use of equation 25 and the valueof the thermal resistance as determined in phase 2.

This approach has several advantages: no temperature change occurs atthe start of the EM experiment, the thermal resistance is determined atthe actual test temperature and the TCR determination is not needed forthe determination of the thermal resistance. Further the current densityexponent n and the activation energy E_(a) can be determined with highaccuracy. This will be detailed subsequently.

As already discussed previously, conventional methods make use of thecross-cut technique to determine the acceleration parameters. Due to theinherent statistical scattering of the degradation curves, a lot ofsamples are needed to get rid of the statistical effects. Thesestatistical effects can be excluded by using only one sample to completethe determination of the acceleration parameters E_(a) and n. Todetermine the current density exponent the interconnect resistance ismeasured during several current density values. It is very importantthat here also the sample temperature remains the same, therefore theAC-current is used to determine the thermal resistance of theinterconnect at a certain current density. When the thermal resistanceis known, the EM test is performed until the current density is raisedagain (FIG. 12).

A clear benefit of this approach is that no temperature change occurs atthe start of the EM experiment. This is particularly important due tothe fact that the current density exponent is determined by the ratio ofthe resistance change slope at the start of the EM experiment and at theend of the previous EM experiment performed at a lower current. Thisratio is important for defining the velocity of resistance change(v_(R)). $\begin{matrix}{v_{R} = {\frac{\mathbb{d}\left( {\Delta\quad{R/R}} \right)}{\mathbb{d}t}\quad\left( {{in}\quad{ppm}\text{/}s} \right)}} & \left( {{equation}\quad 26} \right)\end{matrix}$The parameter v_(R) is inversely proportional to the time to failure:v _(R) =A′·j ^(n) ·e−E/kT with A′ a constant  (equation 27)The current density exponent n can then easily be determined.

Whereas conventional systems keep the current density constant and raisethe oven temperature in certain steps in order to estimate theactivation energy, an approach according to the present disclosureproposes to measure the resistance versus time while the sampletemperature is raised in certain steps and the current density value iskept constant. An AC current is used to determine the thermal resistanceevery time the sample temperature is raised. When the thermal resistanceis known the EM test is started by applying a DC current until thesample temperature is raised again. During this EM test the oventemperature is adjusted in order to keep the sample temperature at thedesired value.

A clear advantage of this measuring method is that the sampletemperature is stable when the EM experiment is started. This isimportant because the activation energy is determined from the ratio ofthe resistance change slope at the start of the EM experiment and at theend of the previous EM experiment performed at a lower temperature.

1-22. (canceled)
 23. A method for determining time to failurecharacteristics of an on-chip interconnect subject to electromigration,comprising the steps of: providing a test structure, being a parallelconnection of a plurality of such on-chip interconnects, performingresistance measurements on at least two time instances on said teststructure under test conditions, said test conditions being at least twoparticular current densities and at least two particular temperatures,and determining from said resistance measurements estimates of said timeto failure characteristics.
 24. A method as in claim 23, wherein saidtime to failure characteristics comprise the median lifetime of saidon-chip interconnects.
 25. A method as in claim 24, wherein estimates ofthe current density exponent n and of the activation energy E_(a) (ineV) are determined from said resistance measurements, using the Blackmodel, given by${{MT}_{F} = {{Aj}^{- n}{\exp\left( \frac{E_{\square}}{k_{B}T} \right)}}},$where MT_(F) denotes the median time to failure of said interconnects ofsaid parallel connection with temperature T (in K) and current density j(in MA/cm²), A being a material constant, and k_(B) being the Boltzmannconstant.
 26. A method as in claim 24, wherein said time to failurecharacteristics further comprise the shape parameter of the time tofailure probability distribution.
 27. A method as in claim 26, furthercomprising a step of correcting via a predetermined relationship saidestimation of said shape parameter, said shape parameter beingdetermined by fitting.
 28. A method as in claim 23, wherein saidparallel connection of said on-chip interconnects is within a singlepackage.
 29. A method as in claim 25, wherein the determining stepcomprises the step of determining said time to failure characteristicsunder real life conditions by extrapolation using said Black model. 30.A method as in claim 23, wherein said measurements are performed at atleast three different temperatures.
 31. A method as in claim 23, whereinsaid on-chip interconnect is in a 90 nm technology.
 32. A method as inclaim 23, wherein said on-chip interconnect is in a sub-90 nmtechnology.
 33. A computer program product, executable on a programmabledevice, containing instructions, which, when executed, perform themethod of: performing resistance measurements on at least two timeinstances on a test structure under test conditions, wherein the teststructure is a parallel connection of a plurality of on-chipinterconnects subject to electromigration, said test conditions being atleast two particular current densities and at least two particulartemperatures, and determining from said resistance measurementsestimates of said time to failure characteristics.